We measured the cross section and response functions (RL , RT , and RLT ) for the 16 O(e, e′ p) reaction in quasielastic kinematics for missing energies 25 ≤ Emiss ≤ 120 MeV at various missing momenta Pmiss ≤ 340 MeV/c. For 25 < Emiss < 50 MeV and Pmiss ≈ 60 MeV/c, the reaction is dominated by single-nucleon knockout from the 1s1/2 -state. At larger Pmiss , the single-particle aspects are increasingly masked by more complicated processes. For Emiss > 60 MeV and Pmiss > 200 MeV/c, the cross section is relatively constant. Calculations which include contributions from pion exchange currents, isobar currents and short-range correlations account for the shape and the transversity but only for half of the magnitude of the measured cross section. PACS numbers: 25.30.Fj, 27.20.+n

The (e, e′ p) reaction in quasielastic kinematics (ω ≈ Q /2mp )1 has long been a useful tool for the study of nuclear structure. (e, e′ p) cross section measurements have provided both a wealth of information on the wave function of protons inside the nucleus and stringent tests of nuclear theories. Response function measurements have provided detailed information about the di?erent reaction mechanisms contributing to the cross section. In the ?rst Born approximation, the unpolarized (e, e′ p) cross section can be separated into four independent response functions, RL (longitudinal), RT (transverse), RLT (longitudinal-transverse), and RT T (transverse-transverse) [1]. These response functions contain all the information that can be extracted from the hadronic system using the (e, e′ p) reaction. Originally, the quasielastic cross section was attributed entirely to single-particle knockout from the valence states of the nucleus. However, a series of 12 C(e, e′ p) experiments performed at MIT-Bates [2–6] measured much larger cross sections at high missing energy than were expected by single-particle knockout models. 12 C(e, e′ p) response function data reported by Ulmer et al. [2] show a substantial increase in the transverse-longitudinal di?erence, (ST ? SL),2 above the two-nucleon emission threshold. Similar RT /RL enhancement has also been observed2

by Lanen et al. for 6 Li [9], by van der Steenhoven et al. for 12 C [10] and, more recently, by Dutta et al. for 12 C, 56 Fe, and 197 Au [11]. There have been several theoretical attempts [12–14] to explain the continuum strength using two-body knockout models and ?nal-state interactions, but no single model has been able to explain all the data. In this ?rst Je?erson Lab Hall A experiment [15], we studied the 16 O(e, e′ p) reaction in the quasielastic region at Q2 = 0.8 (GeV/c)2 and ω = 439 MeV (|q | ≈ 1 GeV/c). We extracted the RL , RT , and RLT response functions from cross sections measured at several beam energies, electron angles, and proton angles for Pmiss ≤ 340 MeV/c. This paper reports the results for Emiss > 25 MeV; p-shell knock-out region (Emiss < 20 MeV) results from this experiment were reported in [16]. We scattered the ?70 ?A continuous electron beam from a waterfall target [17] with three foils, each ?130 mg/cm2 thick. We detected the scattered electrons and knocked-out protons in the two High Resolution Spectrometers (HRSe and HRSh ). The details of the Hall A experimental setup are given in [18,19]. We measured the 16 O(e, e′ p) cross section at three beam energies, keeping |q | and ω ?xed in order to separate response functions and understand systematic uncertainties. Table I shows the experimental kinematics. The accuracy of a response-function separation depends on precisely matching the values of |q | and ω for di?erent kinematic settings. In order to match |q |, we measured 1 H(e, ep) (also using the waterfall target) with a pinhole collimator in front of the HRSe . The momentum of the detected protons was thus equal to q. We determined the 1 H(e, ep) momentum peak to δp ?4 , allowing us to match δ|q|| to 1.5 × 10?4 p = 1.5 × 10 |q between the di?erent kinematic settings. Throughout the experiment, 1 H(e, e) data, measured simultaneously with 16 O(e, e′ p), provided a continuous monitor of both luminosity and beam energy. The radiative corrections to the measured cross sections were performed by two independent methods; using the code RADCOR [19,20], which unfolds the radiative tails in (Emiss , Pmiss ) space, and using the code MCEEP [21] which simulates the radiative tail based on the prescription of Borie and Drechsel [22]. The corrected cross sections from the two methods agreed within the statistical uncertainties of these data. The radiative correction to the continuum cross section for 60 < Emiss < 120 MeV was about 10% of the measured cross section. At θpq = ±8? , RLT extracted independently at beam energies of 1.643 GeV and 2.442 GeV agree well within statistical uncertainties. This indicates that the systematic uncertainties are smaller than the statistical uncertainties. The systematic uncertainty in cross section measurements is about 5%. This uncertainty is dominated by

2

the uncertainty in the 1 H(e, e) cross section to which the data were normalized [23]. Figure 1 shows the measured cross section as a function of missing energy at Ebeam = 2.4 GeV for various proton angles, 2.5? ≤ θpq ≤ 20? . The average missing momentum increases with θpq from 50 MeV/c to 340 MeV/c. The prominent peaks at 12 MeV and 18 MeV are due to 1p-shell proton knockout and are described in [16], where it was shown that the p-shell cross sections can be explained up to Pmiss = 340 MeV/c by relativistic Distorted Wave Impulse Approximation (DWIA) calculations. However the spectra for Emiss > 20 MeV exhibit a very di?erent behavior. At the lowest missing momentum, Pmiss ≈ 50 MeV/c, the wide peak centered at Emiss ≈ 40 MeV is due predominantly to knockout of protons from the 1s1/2 -state. This peak is less prominent at Pmiss ≈ 145 MeV/c and has vanished beneath a ?at background for Pmiss ≥ 200 MeV/c. At Emiss > 60 MeV or Pmiss > 200 MeV/c, the cross section does not depend on Emiss and decreases only weakly with Pmiss . We compared our results to single-particle knockout calculations by Kelly [24] and Ryckebusch [25–27] to determine how much of the observed continuum (Emiss > 20 MeV) cross section can be explained by 1s1/2 -state knockout. Kelly [24] performed DWIA calculations using a relativized Schr¨dinger equation in which the dynamio cal enhancement of lower components of Dirac spinors is represented by an e?ective current operator [28]. These calculations accurately describe the 1p-shell missing momentum distributions up to 340 MeV/c [16]. For the 1s1/2 -state, Kelly used a normalization factor of 0.73 and spread the cross section and the response functions over missing energy using the Lorentzian parameterization of Mahaux [29]. At small Pmiss , where there is a clear peak at 40 MeV, this model describes the data well. At larger Pmiss , where there is no peak at 40 MeV, the DWIA cross section is much smaller than the measured cross section (see Figure 1). Relativistic DWIA calculations by other authors [30,31] show similar results. This con?rms the attribution of the large missing momentum cross section to non-single-nucleon knockout. Figure 1 also shows calculations by Ryckebusch et al. [25–27] using a non-relativistic single-nucleon knockout Hartree-Fock (HF) model which uses the same potential for both the ejectile and bound nucleons. Unlike DWIA, this approach conserves current at the one-body level, but it also requires much smaller normalization factors because it lacks a mechanism for diversion of ?ux from the single-nucleon knockout channel. At small missing momentum, this model describes both the p-shell and s-shell cross sections well. As the missing momentum increases, it progressively overestimates the p-shell and s-shell cross sections. The most important di?erence between the DWIA and HF single-nucleon knockout models is the absorptive potential; its omission from the HF model increases the HF cross section for Pmiss ≈ 300 3

MeV/c by an order of magnitude for both p-shell and s-shell. Figure 1 also shows (e, e′ pn) and (e, e′ pp) contributions to the (e, e′ p) cross section calculated by Ryckebusch etal. [32]. This calculation has also been performed in a HF framework. The cross section for the two particle knock-out has been calculated in the “spectator approximation” assuming that the two knockedout nucleons will escape from the residual A ? 2 system without being subject to inelastic collisions with other nucleons. This calculation includes contributions mediated by pion-exchange currents, intermediate ? creation and central and tensor short-range correlations. According to this calculation, in our kinematics, twobody currents (pion-exchange and ?) account for approximately 85% of the calculated (e, e′ pn) and (e, e′ pp) strength. Short-range tensor correlations contribute approximately 13% while short-range central correlations contribute only about 2%. Since the two-body current contributions are predominantly transverse, the calculated (e, e′ pn) and (e, e′ pp) cross section is mainly transverse in our kinematics. The ?at cross section predicted by this calculation for Emiss > 50 MeV is consistent with the data, but it accounts for only about half the measured cross section. Hence, additional contributions to the cross section such as heavier meson exchange and processes involving more than two hadrons must be considered. Figures 2-4 present the separated response functions for various proton angles. Due to kinematic constraints, we were only able to separate the responses for Emiss < 60 MeV. The separated response functions can be used to check the reaction mechanism. If the excess continuum strength at high Pmiss is dominated by two body processes rather than by correlations, then it should be predominantly transverse. Figure 2 presents the separated response functions for Pmiss ≈ 60 MeV/c. The wide peak centered around Emiss ≈ 40 MeV in both RL and RT corresponds primarily to single-particle knockout from the 1s1/2 -state. The di?erence between the transverse and longitudinal spectral functions (ST ? SL ), which is expected to be zero for a free nucleon, appears to increase slightly with Emiss . The magnitude of (ST ?SL ) measured here is consistent with the decrease in (ST ?SL ) with Q2 seen in the measurements of Ulmer et al. [2] at Q2 = 0.14 (GeV/c)2 and by Dutta et al. [11] at Q2 = 0.6 and 1.8 (GeV/c)2 . This suggests that, in parallel kinematics, transverse nonsingle-nucleon knockout processes decrease with Q2 . Figure 3 presents the separated response functions (RL+T T 3 , RT , and RLT ) for |θpq | = 8? ( Pmiss ≈ 145

3

RL+T T ≡ RL +

VT T VL

RT T

MeV/c). The Mahaux parameterization does not reproduce the shape of RL or of RT as a function of missing energy. For Emiss < 40 MeV, all calculated response functions underestimate the data suggesting the excitation of states with a complex structure between the pand s-shells. For Emiss > 50 MeV, RL+T T (which is T mainly longitudinal because VTL RT T is estimated to be V only about 7% of RL [24] in these kinematics) is consistent with both zero and with the calculations. RT , on the other hand, remains nonzero to at least 60 MeV. RT is also signi?cantly larger than the DWIA calculation. RLT is about twice as large as the DWIA calculation over the entire range of Emiss . RLT is nonzero for Emiss > 50 MeV, indicating that RL is also nonzero in that range. Figure 4 presents the separated response functions for |θpq | = 16? ( Pmiss ≈ 280 MeV/c). At this missing momentum, none of the measured response functions show a peak at Emiss ≈ 40 MeV where single-particle knockout from the 1s1/2 -state is expected. RL+T T is close to zero and the DWIA calculation. However, RT and RLT are much larger than the DWIA calculation. RT is also much larger than RLT indicating that the cross section is due in large part to transverse two-body currents. The fact that RLT is nonzero indicates that RL , although too small to measure directly, is also nonzero. To summarize, we have measured the cross section and response functions (RL , RT , and RLT ) for the 16 O(e, e′ p) reaction in quasielastic kinematics at Q2 = 0.8 (GeV/c)2 and ω = 439 MeV for missing energies 25 < Emiss < 120 MeV at various missing momenta Pmiss ≤ 340 MeV/c. For 25 < Emiss < 50 MeV and Pmiss ≈ 60 MeV/c the reaction is dominated by single-nucleon knockout from the 1s1/2 -state and is described well by DWIA calculations. (ST ? SL ) is smaller than that measured at Q2 = 0.14 [2] and Q2 = 0.6 (GeV/c)2 , but larger than that measured at Q2 = 1.8 (GeV/c)2 [11]. This is consistent with the previous observation that, at low Pmiss , knockout processes due to MEC and IC decrease with Q2 [11]. At increasing missing momenta, the importance of the single-particle aspects is diminished. The cross section and the response functions no longer peak at the maximum of the s-shell (40 MeV). They no longer have the expected Lorentzian shape for s-shell knockout. DWIA calculations underestimate the cross section and response functions at Pmiss > 200 MeV/c by more than a factor of 10. Hence, we conclude that the single-particle aspect of the 1s1/2 -state contributes less than 10% to the cross section at Pmiss > 200 MeV/c. This is in contrast to the p-shell case, where DWIA calculations describe the data well up to Pmiss = 340 MeV/c. At 25 < Emiss < 120 and Pmiss > 200 MeV/c the cross section is almost constant in missing energy and missing momentum. For Emiss > 60 MeV this feature is well reproduced by two-nucleon knockout calculations, (e, e′ pp) plus (e, e′ pn). These calculations also account for the predominantly transverse nature of the cross section, due to 4

the large contribution from the two-body (pion exchange and isobar) currents. This indicates that the excess continuum strength at high Pmiss is dominated by two body processes rather than by correlations. To our knowledge, this is the only model which can account for the shape, transversity and about the half of the magnitude of the measured continuum cross section. The unaccounted for strength suggests that additional currents and processes play an important role. We acknowledge the outstanding support of the sta? of the Accelerator and Physics Divisions at Je?erson Laboratory that made this experiment successful. We thank Dr. J. Ryckebusch for providing theoretical calculations. We also thank Dr. J.M. Udias for providing us with the NLSH bound-state wave functions. This work was supported in part by the U.S. Department of Energy contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Je?erson National Accelerator Facility, other Department of Energy contracts, the National Science Foundation, the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Atomic Energy Commission and National Center of Scienti?c Research, and the Natural Sciences and Engineering Research Council of Canada.

FIG. 1. Cross sections measured at di?erent outgoing proton angles as a function of missing energy. The curves show the single-particle strength calculated by Kelly (solid curve, only s-shell is shown) and by Ryckebusch (dashed curve), folded with the Lorentzian parameterization of Mahaux. The dotted line shows the Ryckebusch et al. calculations of the (e, e′ pn) and (e, e′ pp) contributions to (e, e′ p) including meson-exchange currents (MEC), intermediate ? creation (IC) and central correlations, while the dot-dashed line also includes tensor correlations.

Em (MeV)

5

RT (fm MeV )

RT(fm3MeV-1) RL+TT (fm3MeV-1) RLT (fm3MeV-1)

-1

0.03 0.02 0.01

0.6 0.4 0.2

3

RL (fm MeV )

-1

0 0.2 0.1 0

0 0.03 0.02 0.01 0

3

-0.1

0

MeV )

-4

40 20 0 -20

-0.0025 -0.005

ST-SL (10

-10

-0.0075 -0.01 25 30 35 40 45 50 55 60

25

30

35

40

45

50

55

60

FIG. 2. The separated response functions and the di?erence of the longitudinal and transverse spectral functions for Pmiss ≈ 60 MeV/c. The calculations have been folded with the Lorentzian parameterization of Mahaux and have been binned in the same manner as the data.